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Exponential integrals give closed-form solutions to a large class of commonly occurring transcendental integrals that cannot be evaluated using elementary functions. Integrals of this type include those with an integrand of the form \(t^a e^{t}\) or \(e^{-x^2}\), the latter giving rise to the Gaussian (or normal) probability distribution.

The most general function in this section is the incomplete gamma function, to which all others can be reduced. The incomplete gamma function, in turn, can be expressed using hypergeometric functions (see Hypergeometric functions).

The exponential integral is closely related to the logarithmic
integral. See li() for additional information.

The exponential integral is related to the hyperbolic
and trigonometric integrals (see chi(), shi(),
ci(), si()) similarly to how the ordinary
exponential function is related to the hyperbolic and
trigonometric functions:

The prime number theorem states that the number of primes less
than \(x\) is asymptotic to \(\mathrm{Li}(x)\) (equivalently
\(\mathrm{li}(x)\)). For example, it is known that there are
exactly 1,925,320,391,606,803,968,923 prime numbers less than
\(10^{23}\) [1]. The logarithmic integral provides a very
accurate estimate: